Li-Zhi Liu

Identification of gene regulatory networks from time course gene expression data

Several methods have been proposed to infer gene regulatory networks from time course gene expression data. As the number of genes is much larger than the number of time points at which gene expression (mRNA concentration) is measured, most existing methods need some ad hoc assumptions to infer a unique gene regulatory network from time course gene expression data. It is well known that gene regulatory networks are sparse and stable. However, inferred network from most existing methods may not be stable. In this paper we propose a method to infer sparse and stable gene regulatory networks from time course gene expression data. Instead of ad hoc assumption, we formulate the inference of sparse and stable gene regulatory networks as constraint optimization problems, which can be easily solved. To investigate the performance of our proposed method, computational experiments are conducted on synthetic datasets.

M-matrix-based stability conditions for genetic regulatory networks with time-varying delays and noise perturbations

Stability is essential for designing and controlling any dynamic systems. Recently, the stability of genetic regulatory networks has been widely studied by employing linear matrix inequality (LMI) approach, which results in checking the existence of feasible solutions to high-dimensional LMIs. In the previous study, the authors present several stability conditions for genetic regulatory networks with time-varying delays, based on M-matrix theory and using the non-smooth Lyapunov function, which results in determining whether a low-dimensional matrix is a non-singular M-matrix. However, the previous approach cannot be applied to analyse the stability of genetic regulatory networks with noise perturbations. Here, the authors design a smooth Lyapunov function quadratic in state variables and employ M-matrix theory to derive new stability conditions for genetic regulatory networks with time-varying delays. Theoretically, these conditions are less conservative than existing ones in some genetic regulatory networks. Then the results are extended to genetic regulatory networks with time-varying delays and noise perturbations. For genetic regulatory networks with n genes and n proteins, the derived conditions are to check if an n × n matrix is a non-singular M-matrix. To further present the new theories proposed in this study, three example regulatory networks are analysed.

Estimating parameters of S-systems by an auxiliary function guided coordinate descent method

The S-system, a set of nonlinear ordinary differential equations and derived from the generalized mass action law, is an effective model to describe various biological systems. Parameters in S-systems have significant biological meanings, yet difficult to be estimated because of the nonlinearity and complexity of the model. Given time series biological data, its parameter estimation turns out to be a nonlinear optimization problem. A novel method, auxiliary function guided coordinate descent, is proposed in this paper to solve the optimization problem by cyclically optimizing every parameter. In each iteration, only one parameter value is updated and it proves that the objective function keeps nonincreasing during the iterations. The updating rules in each iteration is simple and efficient. Based on this idea, two algorithms are developed to estimate the S-systems for two different constraint situations. The performances of algorithms are studied in several simulation examples. The results demonstrate the effectiveness of the proposed method.

Reverse engineering of gene regulatory networks from biological data

Reverse engineering of gene regulatory networks (GRNs) is one of the most challenging tasks in systems biology and bioinformatics. It aims at revealing network topologies and regulation relationships between components from biological data. Owing to the development of biotechnologies, various types of biological data are collected from experiments. With the availability of these data, many methods have been developed to infer GRNs. This paper firstly provides an introduction to the basic biological background and the general idea of GRN inferences. Then, different methods are surveyed from two aspects: models that those methods are based on and inference algorithms that those methods use. The advantages and disadvantages of these models and algorithms are discussed.

Properties of sparse penalties on inferring gene regulatory networks from time-course gene expression data

Genes regulate each other and form a gene regulatory network (GRN) to realise biological functions. Elucidating GRN from experimental data remains a challenging problem in systems biology. Numerous techniques have been developed and sparse linear regression methods become a promising approach to infer accurate GRNs. However, most linear methods are either based on steady-state gene expression data or their statistical properties are not analysed. Here, two sparse penalties, adaptive least absolute shrinkage and selection operator and smoothly clipped absolute deviation, are proposed to infer GRNs from time-course gene expression data based on an auto-regressive model and their Oracle properties are proved under mild conditions. The effectiveness of those methods is demonstrated by applications to in silico and real biological data.

Alternating weighted least squares parameter estimation for biological S-systems

The S-system, which is a set of nonlinear ordinary differential equations and derived from the generalized mass action law, is a consistent model to describe various biological systems. Parameters in S-systems contain important biological information and yet can not be obtained directly from experiments. Therefore, the parameter estimation methods are a choice to estimate parameters in S-systems. However, the parameter estimation for this model turns out to be a complex nonlinear optimization problem. A novel method, alternating weighted least squares (AWLS), is proposed in this paper to estimate the parameters in S-systems. The fast deterministic AWLS method takes advantage of the special structure of the S-system model and reduces solving the nonlinear optimization problem into alternately solving weighed least squares problems which have analytical solutions. The effectiveness of AWLS is demonstrated by the simulation studies and the results show that the AWLS outperforms the existing alternating regression method.

Nonlinear Model-Based Method for Clustering Periodically Expressed Genes

Clustering periodically expressed genes from their time-course expression data could help understand the molecular mechanism of those biological processes. In this paper, we propose a nonlinear model-based clustering method for periodically expressed gene profiles. As periodically expressed genes are associated with periodic biological processes, the proposed method naturally assumes that a periodically expressed gene dataset is generated by a number of periodical processes. Each periodical process is modelled by a linear combination of trigonometric sine and cosine functions in time plus a Gaussian noise term. A two stage method is proposed to estimate the model parameter, and a relocation-iteration algorithm is employed to assign each gene to an appropriate cluster. A bootstrapping method and an average adjusted Rand index (AARI) are employed to measure the quality of clustering. One synthetic dataset and two biological datasets were employed to evaluate the performance of the proposed method. The results show that our method allows the better quality clustering than other clustering methods (e.g., k-means) for periodically expressed gene data, and thus it is an effective cluster analysis method for periodically expressed gene data.

Structure identification and parameter estimation of biological s-systems

Reconstruction of a biological system from its experimental time series data is a challenging task in systems biology. The S-system which consists of a group of nonlinear ordinary differential equations is an effective model to characterize molecular biological systems and analyze the system dynamics. However, inference of S-systems without the knowledge of system structure is not a trivial task due to its nonlinearity and complexity. In this paper, a pruning separable parameter estimation algorithm is proposed for inferring S-systems. This novel algorithm combines the separable parameter estimation method and a pruning strategy, which includes adding an ℓ1 regularization term to the objective function and pruning the solution with a threshold value. The performance of the pruning strategy in the proposed algorithm is evaluated from two aspects: the parameter estimation error and structure identification accuracy. The proposed algorithm is applied to two S-systems with simulated data. The results show that the proposed algorithm has much lower estimation error and much higher identification accuracy than the existing method.

Robust inference of gene regulatory networks from multiple microarray datasets

Multiple time-course microarray datasets with the same underlying gene network are collected from different experiments. The inference of gene regulatory networks (GRNs) can be improved by integrating these datasets. Microarray data may be contaminated with large errors or outliers, which may affect the inference results. A novel method, Huber group LASSO, is proposed to reconstruct the GRNs from multiple datasets as well as taking the robustness into account. To solve the optimization problem involved in the proposed method, an efficient algorithm which combines the ideas of auxiliary function minimization and block coordinate descent is developed. Simulations and real data applications demonstrate the effectiveness of our method. Results show that the proposed method outperforms the group LASSO method and is able to reconstruct reasonably good GRNs from multiple datasets even the number of genes exceeds the number of observations.

Parameter estimation method for improper fractional models and its application to molecular biological systems

Derived from biochemical principles, molecular biological systems can be described by a group of differential equations. Generally these differential equations contain fractional functions plus polynomials (which we call improper fractional model) as reaction rates. As a result, molecular biological systems are nonlinear in both parameters and states. It is well known that it is challenging to estimate parameters nonlinear in a model. However, in fractional functions both the denominator and numerator are linear in the parameters while polynomials are also linear in parameters. Based on this observation, we develop an iterative linear least squares method for estimating parameters in biological systems modeled by improper fractional functions. The basic idea is to transfer optimizing a nonlinear least squares objective function into iteratively solving a sequence of linear least squares problems. The developed method is applied to the estimation of parameters in a metabolism system. The simulation results show the superior performance of the proposed method for estimating parameters in such molecular biological systems.